Breakdown and reconstruction of figural concepts in proofs by contradiction in geometry

In this paper we aim to contribute to the discussion on the role of images in respect to proof, elaborating on two examples of indirect arguments produced by students. The analysis is framed by the construct of figural concepts (Fischbein, 1993) and by the model for indirect proof introduced in (Antonini & Mariotti, 2008). According to the general assumption that the harmony between the figural and the conceptual aspect is required for productive reasoning, we will present two cases where the production of an indirect argument can be interpreted as emerging from the need of restoring such harmony. In particular, the quest for an image after the contradiction seems to be crucial to support the passage from the ‘absurd’ to the validation of the original statement.